The Spaniard prodded home a loose ball from close range in the 93rd minute to complete a second half fightback and secure our 11th consecutive victory in the competition.
City’s lead at the top of the table could be cut when second-placed Chelsea face third-placed Liverpool on Sunday and it looked as if we were destined to leave North London with a draw that would have boosted our title rivals ahead of their meeting at Stamford Bridge.
Pep Guardiola’s side fell behind when Bukayo Saka swept home Kieran Tierney’s pass to finish off a slick passing move during a first half in which the Gunners were in the ascendancy.
However, the game was turned upside down during seven chaotic second half minutes, when Riyad Mahrez converted a 57th minute penalty to level things up.
The spot kick was awarded after Bernardo Silva tumbled over Granit Xhaka’s outstretched leg.
Referee Stuart Attwell waved away City’s appeals, but overturned his decision after consulting the pitch-side monitor and seeing that Xhaka had also pulled the midfielder’s shirt.
Mahrez sent Aaron Ramsdale the wrong way from a penalty spot which had been scuffed by Gabriel during the wait for the decision, an act for which he was booked, and it proved costly moments later when the Brazilian body checked Gabriel Jesus and was dismissed with a second yellow card.
A minute prior, the impressive Gabriel Martinelli had wasted a golden opportunity, cracking the outside of the post with the goal gaping after Nathan Ake had been forced into a goal line clearance following a mix up between Aymeric Laporte and Ederson.
Arsenal had their own appeal for a penalty turned down in the first half when VAR supported Attwell’s decision to award a corner after Ederson collided with Martin Odegaard as he cleared the ball.
City dig deep
Before the game, the Catalan declared City would face the best Arsenal team he has encountered during his time in England.
Like City, they came into the contest in good form having won their last five in all competitions and, despite the absence of manager Mikel Arteta due to a positive COVID-19 test, it was the hosts who started the brighter.
City had a couple of early chances. Gabriel Jesus planted a header the wrong side of the post, whilst Ruben Dias spurned a glorious opportunity when he glanced Raheem Sterling’s cross wide, but Arsenal played with intensity and looked the more dangerous of the two teams before the break.
Martinelli was their talisman. He forced Ederson into a smart stop down to his left, curled an effort from distance inches over the bar and poked wide from a tight angle following a mazy run past Dias and Joao Cancelo.
Their early dominance paid off as they took the lead in style, moving the ball through midfield before Tierney picked out Saka for a composed first-time finish, but it was undone once Gabriel received his marching orders.
City were on top against the 10 men, with Mahrez firing into the side netting as we pushed for a winner, but Arsenal's hard work looked to have earned them a point until Rodrigo popped up to complete a memorable fight back which feels like a key moment in the season so far.
Pep guardiola reaction
"Arsenal were better. We face a team that in recent years has struggled to be in the top four. They start this season bottom.
"They have six days after beating Norwich, we have two days. It was so tough for us to come back. We had no energy. That is why we put one more player in the middle to get more control and passes.
"We know in transition and the physicality they have it can be difficult and it was difficult. We tried and in one minute we have a penalty and then the red card. We try to analyse many things but sometimes the coin falls on your side."
Man of the Match: Nathan Ake
With Laporte heading over Ederson as he came to claim the ball, the Frenchman looked certain to score an own goal, but Ake showed incredible desire to get back and hook the ball off the line just in time.
A crucial intervention which was as valuable as a goal.
Rodri's strike, timed at 92:28, is our latest winning goal in a Premier League match since Gabriel Jesus scored on the final day of the 2017-18 season against Southampton to ensure Guardiola's men became the first team to reach 100 points.
What it means
City begin 2022 with an 11th successive Premier League victory that moves us 11 points clear at the top of the table.
In the competition’s history, only Manchester United in 1993-94 and City in 2017-18 (both 12 points) had a bigger lead at the end of New Year’s Day.
Should Chelsea beat Liverpool on Sunday, our advantage will be cut to eight points, whilst a win for the Reds – who have a game in hand - would reduce our lead to nine points.
ARSENAL: Ramsdale, Tomiyasu, White, Gabriel, Tierney, Partey, Xhaka, Odegaard (Holding 63’), Saka (Elneny 84’), Martinelli, Lacazette (Smith Rowe).
SUBS: Leno, Maitland-Niles, Soares, Pepe, Tavares, Lokonga.
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